Theorem expandral 40700

Description: Expand a restricted universal quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypothesis

Ref	Expression
expandral.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
expandral	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))

Proof of Theorem expandral

Step	Hyp	Ref	Expression
1		expandral.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	ralbii 3164	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)
3		df-ral 3142	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4	2, 3	bitri 277	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1534   ∈ wcel 2113  ∀wral 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 209  df-ral 3142

This theorem is referenced by:  expanduniss  40703  ismnuprim  40704